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If I understand correctly, it is actually easier to remove the second electron. However there are cases when the second electron fills the orbital completely and therefore a higher ionization energy is needed in order to remove the electron.
Removing the second electron is more difficult because the atom pulls on the second electron stronger. In the example he gave us, after removing one electron from copper, it became Cu+, therefore the nuclear charge is stronger than Cu.
The second electron is more difficult to remove because after the first electron is removed, the atom has a positive net charge. This means that there are now more protons in the atom than electrons so there is a stronger attraction between the positive nucleus and the remaining electrons. This increased attraction makes it harder to remove any of the remaining electrons.
Removing the second electron is harder because as stated in class the atom becomes a cation (which normally attract electrons from other atoms) so it pulls on the electron more than before through a high Zeffective.
Each removal of a valence electron will take more energy than the last and be more difficult. This is due to a concept where it will take more energy to remove an electron from an atom that already has a positive charge because of a higher electrostatic force.
I think you can try to think it this way: the electron on the outer shells are generally less stable than those on the more inside shells, because the inner electrons are closer to the protons so that there is stronger electromagnetic attraction. Removing a second electron is to remove one inner electron compared to the first electron moved, so it requires higher energy.
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